I would go for B.
Bill Walton continued_______to his position as a regular starter in the game he loved.
'...continued...' should be followed by '...playing...', the use of '...on playing...' is incorrect.
now we are left with option B and C.
C is wrong because:
(C) playing, though injured over and over
, and he was always hoping to return back
'...to play...' could be used but the last part of the option does not link well with the remaining sentence
(E) to play, despite recurring injuries, hoping that the return
...to his position...'
"Always....Read between the lines"